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Powers of complex numbers

Sal simplifies the 20th power of a complex number given in polar form. Created by Sal Khan.

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• Where do I find the derivation of Euler's formula as I definitely don't remember e^itheta = cos etc etc! Have I missed this earlier on?
• It's confusing because this video appears in pre-calculus, before learning euler's formula.. I think I'm going to skip the complex number stuff for now until I hit it in calculus!
• Why not use de Moivre's Theorem instead? We could just do: (cos(2π/3)+i sin(2π))^20 =
(cos(40π/3)+ i sin (40π/3)) which seems a little faster.
• yeah, i would do it that way, too, but i guess it's good to learn different methods
• I have used a calculator to caculate 2^i. It gave me this irational complex number of approximately .7692389014 + .6389612763i

I was expecting a rational complex number(One that can be expressed as the sum of 2 fractions) as the result since I was taking an integer to a non-irrational power.

How come when I take anything to the i power I get an irrational and in fact transcendental complex number when neither the base nor the exponent are irrational?

I mean I shouldn't get irrational = rational^rational unless it is a fractional exponent and ^i is not a fractional exponent, not even a complex one.
• The properties involving i are a bit different. But,
a^i = cos(ln a)+i sin(ln a)
Therefore,
2^i = cos ( ln 2 ) + i sin( ln 2 )
• Why did sal count the angle 4/3pi again from the beginning angle i.e 0 pi. He should be starting from 2/3 pi right?
• I dont know exactly at what time he said that or what time you mean.

But at around he has 13 1/3 pi. This is the same as 12pi + 4/3 pi.
So with the 12pi you go around the circle 6 times (2pi = going around 1 time).
Then you have 4/3 pi left.
• Would i be correct in saying that; a scalar in front of the exponential form would increase the modulus of the complex number as well as increase the angle?
• If there is a scalar applied to a complex number raised to an exponent, the exponent would apply to both the scalar and the original number.
• how is any angle equal to 2 pi k + the angle?
• Those angles are not equal, but they put you in the same location in the plane.
• Is there any way to do this if the complex number is in rectangular form?
• Well sure, you can use binomial theorem and expand the power. For even powers, you can first square the complex number, and then take that result to half the original power which can be quick depending on the complex number and the exponent.
But using exponential form and de'Moivre is a lot easier and less time consuming.
• Hi, I am working on a problem with the following introduction:

`Find the solution of the following equation whose argument is strictly between 180 degrees and 270 degrees.z^5=-243iz=?`

There is an explanation regarding solving for theta, as follows:

`Remember that theta is strictly between 180 and 270 degrees. Therefore, we need to find the multiple of 72 degrees that is strictly within the range of 180−54 degrees = 126 degrees, and 270-54 degrees = 216 degrees. This multiple is simply 144 degree, so theta equals 198 degree.`

To my mind it is not entirely clear that 216 isn't also a multiple of 72 in that range. Is this my mistake? Do "between" and "within" strictly mean greater and less than but not equal to?

Doing so still gives me z such that z^5 = -243i but it is apparently incorrect.
• I'm running into the same issue. Did you figure this out?
• I don't understan, i wish they can redo this